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<p><it>Abstract</it>—The problem of designing space-optimal 2D regular <it>N</it>×<it>N</it>×<it>N</it> cubical mesh algorithms with linear schedule <it>ai</it>+<it>bj</it>+<it>ck</it>, 1 ≤<it>a</it>≤<it>b</it>≤<it>c</it>, and <it>N</it>=<it>nc</it>, is studied. Three novel nonlinear processor allocation methods, each of which works by combining a partitioning technique (<it>gcd-partition</it>) with different nonlinear processor allocation procedures (<it>traces</it>), are proposed to handle different cases. In cases where <it>a</it>+<it>b</it>≤<it>c</it>, which are dealt with by the first processor allocation method, space-optimal designs can always be obtained in which the number of processing elements is equal to <math><tmath>${N^2\over c}$</tmath></math>. For other cases where <it>a</it>+<it>b</it> > <it>c</it> and either <it>a</it>=<it>b</it> and <it>b</it>=<it>c</it>, two other optimal processor allocation methods are proposed. Besides, the closed form expressions for the optimal number of processing elements are derived for these cases.</p>
Algorithm mapping, data dependency, linear schedule, matrix multiplication, optimizing compiler, space-optimal, systolic array.

J. Tsay and P. Chang, "Design of Space-Optimal Regular Arrays for Algorithms with Linear Schedules," in IEEE Transactions on Computers, vol. 44, no. , pp. 683-694, 1995.
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